98 



HYDRAULICS AND ITS APPLICATIONS 



and radial motion combined in the required proportions, since in each 

 case the velocity is inversely proportional to the radius. The angle 

 between the stream line and the corresponding radius vector at any point 

 will then be constant, and the stream lines will form a series of equi- 

 angular or logarithmic spirals. The difference of pressure between any 

 two points may then be found by adding the pressure differences due to 

 the two methods of flow taken separately. 



Strictly this should only be applied to cases of inward flow, since 

 outward flow causes instability. 



In a free spiral vortex formed under normal conditions, the surface 

 resistances tend to prevent the attainment of such high velocities near 

 the centre of rotation as are indicated by the preceding analysis. In 

 consequence the actual values of C z for small radii are less than those 

 calculated. ^The following observations by the author on the surface 

 profile of such a vortex, produced by the discharge from a circular 

 cylindrical vessel 2 feet in diameter under a constant head of 9 inches, 

 bear out this point. 



Since the velocity varies inversely as the radius, and since this velocity 

 cannot be infinite at the axis where r = o, an air column is essential 

 at the centre of a free vortex. When this air column cannot be main- 

 tained, we get a combination of a forced vortex at and near the axis 

 and a free vortex at points further removed (Fig. 51). 



If a be the radius at which the two surface curves intersect, the depth 

 of the central depression below the general level of the surface may be 



w* a 2 

 shown to be given by - - feet. 



This is termed a Compound Vortex, and the state of affairs there 

 existing is of importance in its application to the theory of the flow of 



